Stochastic flow shop scheduling
In this thesis we present new results for the makespan and the flowtime in a flow shop without intermediate storage between machines. We consider m machines and n jobs with random processing times. Since there is no intermediate storage between machines, a job which has finished processing at one machine may have to stay at that machine until the next machine is free. This phenomenon is known as blocking. Our goal is to select the optimal schedule; in our case, the schedule which in some sense minimizes the makespan or the flowtime. Makespan is the total time required to process a set of jobs and flowtime is sum of all the times at which jobs are completed.
Our results require various stochastic orderings on the processing time distributions. Some of these orderings minimize the expected flowtime or expected makespan, and some stochastically minimize the makespan. The stochastic minimization results are much stronger. The optimum sequence in these cases not only minimize the expected makespan, but also maximize the probability of completing a set of jobs by time t for any t.
Our last result resolves the conjecture of Pinedo (1982a) that in a stochastic flow shop with m machines, n-2 deterministic jobs with unit processing time, and two stochastic jobs each with mean one, the sequence which minimizes the expected makespan has one of the stochastic jobs first and the other last. We prove that Pinedo's conjecture is almost true. We prove that either the sequence suggested by Pinedo or a sequence in which the stochastic jobs are adjacent at one end of the sequence minimizes the expected makespan. Our result does not require the stochastic jobs to have an expected value of one. Furthermore, we show that our result cannot be improved in the sense that in some cases one sequence is strictly optimal and in other cases the other is strictly optimal.